Subgroups — #28378190

Order 13952.

62 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16812749	C₂	1	2	yes	maximal
2	C₁₆×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic
3	#16812749	—	2	2	no	—
4	C₂×#13182004	C₂²	1	1	yes	characteristic
5	C₈×Dic₁₀₉	C₄	1	1	yes	characteristic
6	C₂×C₁₆×C₁₀₉	C₄	1	1	yes	characteristic
7	C₂×#11724784	C₈	1	1	yes	characteristic
8	C₄×Dic₁₀₉	C₈	1	1	yes	characteristic
9	#13182004	D₄	1	1	yes	characteristic
10	#13182004	Q₈	1	1	yes	characteristic
11	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
12	C₁₆×C₁₀₉	—	2	1	no	—
13	C₂×C₄×C₁₀₉	C₂×C₈	1	1	yes	characteristic
14	C₂×Dic₁₀₉	C₁₆	1	1	yes	characteristic
15	C₂×Dic₁₀₉	C₁₆	1	1	yes	characteristic
16	C₈×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
17	C₈×C₁₀₉	M₁₆	1	1	yes	characteristic
18	#11724784	—	2	1	no	—
19	C₂²×C₁₀₉	C₂×C₁₆	1	1	yes	characteristic
20	C₄×C₁₀₉	C₄⋊C₈	1	1	yes	characteristic
21	C₄×C₁₀₉	M₃₂	1	1	yes	characteristic
22	Dic₁₀₉	—	2	1	no	—
23	Dic₁₀₉	C₃₂	1	2	yes	complemented
24	C₂×C₁₀₉	C₄⋊C₁₆	1	1	yes	characteristic
25	C₂×C₁₀₉	C₂×C₃₂	1	1	yes	characteristic
26	C₂×C₁₀₉	M₆₄	1	1	yes	characteristic
27	C₄⋊C₃₂	—	109	1	no	maximal, Sylow
28	C₁₀₉	C₄⋊C₃₂	1	1	yes	characteristic, complemented, Sylow
29	C₂×C₃₂	—	109	2	no	—
30	C₄×C₁₆	—	109	1	no	—
31	C₂×C₁₆	—	109	1	no	—
32	C₂×C₁₆	#89137	1	1	yes	characteristic
33	C₄×C₈	—	109	1	no	—
34	C₃₂	—	218	2	no	—
35	C₂×C₈	—	109	1	no	—
36	C₂×C₈	C₂×#89137	1	1	yes	characteristic
37	C₄²	—	109	1	no	—
38	C₁₆	—	2	1	no	—
39	C₁₆	—	109	1	no	—
40	C₁₆	—	109	1	no	—
41	C₂×C₄	—	109	1	no	—
42	C₂×C₄	—	109	1	no	—
43	C₂×C₄	D₁₀₉⋊C₈	1	1	yes	characteristic
44	C₈	—	218	1	no	—
45	C₈	C₄⋊#89137	1	1	yes	characteristic
46	C₈	#13182038	1	1	yes	characteristic
47	C₂²	D₁₀₉⋊C₁₆	1	1	yes	characteristic
48	C₄	—	109	2	no	—
49	C₄	—	218	1	no	—
50	C₄	#16812766	1	1	yes	characteristic
51	C₄	#16812774	1	1	yes	characteristic
52	C₂	#20295947	1	1	yes	characteristic
53	C₂	D₁₀₉⋊C₃₂	1	1	yes	characteristic
54	C₂	#20295959	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂×#89137
commutator	C₂×C₁₀₉	C₂×C₃₂
Fitting	C₂×C₁₆×C₁₀₉	C₄
socle, prime radical	C₂²×C₁₀₉	C₂×C₁₆
cosocle	C₂×#13182004	C₂²
hypercenter	C₂×C₁₆	#89137
hypocenter	C₁₀₉	C₄⋊C₃₂
LCS	G ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₁₆

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
23. Dic₁₀₉	C₃₂	semidirect
28. C₁₀₉	C₄⋊C₃₂	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#16812749	2
C₁₆×Dic₁₀₉	2
C₄⋊C₃₂	109